Absolute Value Inequalities

Definition

Absolute Value Inequalities are inequalities where a variable or expression is contained within absolute value bars. They are used to describe “neighborhoods” or ranges of values within a certain distance of a target.

• How to read: “Absolute value of u is less than a” or “greater than a.”
• Meaning / when to use: Specifies a range of tolerance or a zone of exclusion.

Why It Matters

These inequalities define “neighborhoods of tolerance” in the physical world. They are the language of precision—allowing us to specify exactly how far a component, signal, or value can deviate from a target before it becomes unusable or dangerous. This is the foundation of Precise Definition of a Limit in calculus.

Core Concepts

• “Less Than” ($|u| < a$): Equivalent to $-a < u < a$ (an “AND” statement). Geometrically, this is the single connected interval centered at zero with radius $a$.
• “Greater Than” ($|u| > a$): Equivalent to $u < -a$ OR $u > a$. Geometrically, this represents everything outside the interval centered at zero, resulting in two separate pieces.
• Radius and Center: $|x - c| < r$ defines an interval of radius $r$ centered at $c$.
• Non-negativity: If $a < 0$, $|u| < a$ has no solution, while $|u| > a$ is true for all $u$.

Connected Concepts

• Absolute Value Equations
• Absolute Value
• Linear Inequalities and Interval Notation
• limit definition precise
• Continuity at a Point